1. Linear Interpolation Applying “weighted averages” to some graphics problems: animations and curve-drawing

2. What does ‘between’ mean? P G B1 B2 The green point G lies between the two blue points B1 and B2. The pink point P does NOT lie between the two blue points B1 and B2.

3. Using a “weighted average” Suppose (x1,y1) and (x2,y2) are points The point located half-way in-between is: midpoint = (½)(x1,y1) + (½)(x2,y2) It’s the “average” of (x1,y1) and (x2,y2) Here’s another point on the line-segment that lies between (x1,y1) and (x2,y2): (x’,y’) = (¼)(x1,y1) + (¾)(x2,y2) It’s a “weighted average” of the endpoints

4. The generalization Let B1 = (x1,y1) and B2 = (x2,y2) be the two endpoints of a line-segment. Suppose w1 and w2 are “weights” (i.e., neither is negative, and their sum equals 1). Then the point P = w1*B1 + w2*B2 is called a “weighted average” of B1 and B2, and P will be located “in-between” B1 and B2. Here P is obtained by “linear interpolation”

5. Describing a line-segment Mathematical description of line-segments Let B1 = (x1,y1) and B2 = (x2,y2) be the two end-points of a given line-segment Let t be a real number whose value can vary continuously, from 0.0 to 1.0 Then point P = (1-t)*B1 + t*B2 will vary continuously over the entire line-segment, starting at B1 (when t=0.0) and ending up at B2 (when t=1.0)

6. Animating a line-segment initial position final position in-between positions

7. The programming idea We only need to specify the segment’s two endpoints at the start and the finish As the segment moves all its intermediate endpoint locations are then calculated as linear interpolations (“weighted averages”) This idea can be simultaneously applied to lots of different line-segments (e.g., to all the sides of a polygon, or to all the “edges” in a wire-frame model)

8. The ‘polyline’ structure typedef struct { float x, y; } point_t; typedef struct { int numverts; point_t vert[ MAXVERT ]; } polyline_t; // declare two polylines (for start and finish) // and a variable polyline (for “in-betweens”) tween[i].x = (1-t)*B1[i].x + t*B2[i].x; tween[i].y = (1-t)*B1[i].y + t*B2[i].y;

9. The ‘tweening.cpp’ demo We illustrate this idea for animating simple polygons, using random-numbers for the coordinates of the starting vertices and the ending vertices We use linear interpolation to calculate the sequence of the “in-between” vertices We use Bresenham’s line-drawing method to “connect-the-dots” at each stage

10. Stick man

11. Drawing curves Another application of “linear interpolation” We can construct a so-called Bezier curve The curve is determined by specifying a small number of “control points” A recursive algorithm is then applied, to generate locations along a smooth curve This idea is ‘deCasteljau’s algorithm’ Kai Long has written an implementation

12. Here’s the idea P1 P2 P3 P4 Start with some “control points” (Here we use just four of them) Find their “weighted averages” Only the red point actually is drawn The same value of t is used for all of these interpolations

13. Kai’s Implementation typedef struct { double h, v; } Point; typedef struct { int numcpts; Point cpts[ MAXVERT ]; } BezierCruve; // helper function void middle( Point p, Point q, Point &mid ) { mid.x = (p.x + q.x)/2; mid.y = (p.y + q.y)/2; }

14. Labels used in recursion P1 p2 h1 h2 a c1 c2 b1 b2 d if ( very_near( p1, p2 ) // base case draw_line_segment( p1, p2 ); else { // recursion case recursive_bezier( p1, b1, c1, d ); recursive_bezier( d, c2, b2, p2 ); }

15. In-class exercise Can you combine these two applications? Create a bezier curve with 4 control-points Create another one with 4 control-points Construct some in-between Bezier curves by applying linear-interpolation to pairs of corresponding control-points So first curve will “morph” into second one